

A266394


Square array a(n,k) is the number of terms in the "continued fraction" of the form k1 + 1/(k2  1/(k3 1/( ... for the fraction k/n.


0



1, 4, 2, 7, 1, 3, 10, 5, 5, 4, 13, 4, 1, 2, 5, 16, 8, 6, 8, 6, 6, 19, 7, 8, 1, 6, 3, 7, 22, 11, 4, 7, 11, 2, 7, 8, 25, 10, 9, 5, 1, 5, 9, 4, 9, 28, 14, 11, 11, 8, 14, 7, 7, 8, 10, 31, 13, 7, 4, 9, 1, 9, 2, 3, 5, 11, 34, 17, 12, 10, 9, 9, 17, 9, 12, 10, 9, 12
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OFFSET

1,2


COMMENTS

a(n,k) is the number of steps to reach 0 for the fraction k/n in the following process: if the fraction f is positive, it is replaced by 1/f; and if it is negative, it is replaced by f+1.


LINKS

Table of n, a(n) for n=1..78.
Maxime Bourrigan, Marie Lhuissier, Enchevêtrements rationnels et autres sorcelleries mathématiques, Images des Mathématiques, CNRS, 2015 (in French).


EXAMPLE

a(1, 3) is the number of steps for 3/1: 3 > 2 > 1 > 0 = 3 steps.
a(3, 1) is the number of steps for 1/3: 1/3 > 2/3 > 3/2 > 1/2 > 1/2 > 2 > 1 > 0 = 7 steps.
The array begins:
1, 2, 3, 4, 5, ...
4, 1, 5, 2, 6, ...
7, 5, 1, 8, 6, ...
10, 4, 6, 1, 11, ...
13, 8, 8, 7, 1, ...
...


PROG

(PARI) trans(f) = if (f > 0, 1/f, if (f < 0, f+1, f));
count(f) = nb = 0; while(f!=0, f = trans(f); nb++); nb;
tabl(nn) = {for (n=1, nn, for (k=1, nn, print1(count(k/n), ", "); ); print(); ); }


CROSSREFS

Cf. A000012 (diagonal), A016777 (1st column), A168230 (2nd line).
Sequence in context: A181051 A299631 A205143 * A286842 A087056 A076129
Adjacent sequences: A266391 A266392 A266393 * A266395 A266396 A266397


KEYWORD

nonn,tabl


AUTHOR

Michel Marcus, Dec 29 2015


STATUS

approved



